Theorem sbcieg 3664

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 10-Nov-2005.)

Hypothesis

Ref	Expression
sbcieg.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbcieg	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

Distinct variable groups:   𝑥,𝐴   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem sbcieg

Step	Hyp	Ref	Expression
1		nfv 2010	. 2 ⊢ Ⅎ𝑥𝜓
2		sbcieg.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	1, 2	sbciegf 3663	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1653   ∈ wcel 2157  [wsbc 3631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-12 2213  ax-ext 2775

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-v 3385  df-sbc 3632

This theorem is referenced by:  sbcie  3666  ralsng  4407  rexsng  4408  rabsnif  4445  ralrnmpt  6592  fpwwe2lem3  9741  nn1suc  11333  opfi1uzind  13528  mrcmndind  17678  fgcl  22007  cfinfil  22022  csdfil  22023  supfil  22024  fin1aufil  22061  ifeqeqx  29871  nn0min  30077  bnj1452  31629  cdlemk35s  36950  cdlemk39s  36952  cdlemk42  36954  2nn0ind  38283  zindbi  38284